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International audience; In current approaches of mathematics education at the upper secondary level, activities proposed to students involve several domains in interaction. After studying activities about modelling or functions, we question here the development in many mathematical curricula around the world of activities involving computer programming, sometimes labelled “coding” or “algorithmics”. The motivation of this paper is that a suitable theoretical framework is required to make sense of students’ work in activities involving various domains, taking into account the semiotic dimension as well as the use of instruments, and the contents and reasoning specific to each domain.

The new programs of French High schools, since 2010, precise objectives in terms of algorithmics. According to High schools curricula, algorithmics teaching appears as a tool (in the sense of Douady, 1986) to give meaning to some studied notions. How to go beyond this level so that algorithmic becomes an object of learning (in the sense of Douady, 1986)? This research work is in the framework of learning of mathematical knowledge in algorithmics at the level of Grade 10 and Scientific Terminal Cycle (Grades 11 and 12) of the French high school. The study and construction of algorithms by students are located in a more general framework of reasoning and proof, but also mathematical modelling. We build three didactic engineerings in High school to study the work of student and to watch teacher’s practices. Our aim is to infer more general results on mathematical reasoning in some specific mathematical domains.The research work favours algorithms’ place in mathematics teaching. We propose a theoretical framework taking into account the general frameworks of mathematics didactics, in particular the Mathematical Working Spaces (MWS) (Kuzniak, Richard, 2014) associated with specific mathematical domains.Following the specification of an Algorithmics Working Spaces (AWS) (Laval, 2014, 2016) we specify the possibilities of the epistemological and cognitive plans inside of these spaces increasing their interactions with their semiotician, instrumental and discursive geneses. We also study which personal spaces can be built for students at different levels of High school system, and how they articulate knowledge about algorithms and school mathematical domains. The models of MWS/AWS aim at analysing of mathematical work in specific mathematical domains, with in particular, paradigms guiding and directing the work of the student.Moreover, since few studies of modelling tasks have been built on MWS/AWS models, we refined some our analyses in the framework MWS/AWS basing on the modelling cycle proposed by Blum & Leiss (2005) in relation to some specific mathematical domains.We build several didactic engineerings that we experimented in various mathematical domains: (1) elementary number theory; (2) mathematical analysis; (3) probabilities and random simulations. These didactic engineerings are experimented and analysed in various French High school's grade: Grade 10 and Scientific Terminal Cycle (Grades 11 and 12). Our research work includes tools for analysing tasks and activities in different mathematical domains. The methodology obtains aggregated global data and finely observes students' activities in classroom and teacher’s practices, Les nouveaux programmes des lycées français, mis en place depuis la rentrée 2010, ont fixé des objectifs précis en matière d’algorithmique. A la lecture de ces programmes, l’enseignement de l’algorithmique apparaît comme outil (au sens de Douady, 1986) pour donner sens à un certain nombre de notions étudiées. Comment dépasser ce stade pour que l’algorithmique devienne objet d’apprentissage (au sens de Douady, 1986) ? Le travail de recherche se situe dans le cadre d’apprentissages de connaissances sur les algorithmes en mathématiques dans l’enseignement au niveau des classes de Seconde et du Cycle Terminal Scientifique du lycée. L’étude et la construction d’algorithmes par les élèves sont situées dans un cadre plus général de raisonnement et de preuve, mais aussi de démarches de modélisation en mathématiques. Il s’agit d’étudier l’effectivité de tels enseignements dans le cadre institutionnel français du point de vue des apprentissages effectivement réalisés par les élèves et des pratiques des enseignants, et d’en inférer des résultats plus généraux sur le raisonnement mathématique dans certains domaines spécifiques, pour les classes du lycée. Le travail de recherche entrepris privilégie la place occupée par les algorithmes dans l’enseignement des mathématiques et propose un cadre théorique tenant compte des cadres généraux de la didactique des mathématiques, en particulier les Espaces de Travail Mathématique (ETM) (Kuzniak, Richard, 2014) associés à des domaines mathématiques spécifiques. Plus particulièrement, poursuivant la spécification d’un modèle Espaces de Travail Algorithmique (ETA) (Laval, 2014, 2016), nous précisons ce que peuvent être les plans épistémologique et cognitif dans ces espaces en mettant l’accent sur leurs interactions liées aux genèses sémiotique, instrumentale et discursive auxquelles ces plans donnent lieu. Nous étudions aussi quels espaces personnels peuvent se construire chez les élèves des différents niveaux scolaires du lycée, et comment ils articulent des connaissances sur les algorithmes et les domaines mathématiques scolaires. Les modèles des ETM/ETA sont consacrés à l’analyse du travail mathématique dans des domaines mathématiques spécifiques avec, en particulier, des paradigmes guidant et orientant le travail des élèves. De plus, partant du fait que peu d’études sur des tâches de modélisation ont été basées sur les modèles ETM/ETA, nous affinons certaines de nos analyses dans le cadre des ETM/ETA sur la base du cycle de modélisation proposé par Blum et Leiss (2005) en relation avec certains domaines spécifiques des mathématiques. Pour cela, nous construisons plusieurs ingénieries didactiques mettant en place des expérimentations dans trois domaines mathématiques : (1) la théorie élémentaire des nombres ; (2) l’analyse ; (3) les probabilités et les simulations aléatoires. Ces ingénieries sont expérimentées et analysées dans les trois niveaux du lycée français : seconde et cycle terminal scientifique. Notre travail de recherche comporte des outils d’analyse de tâches et d’activités dans différents domaines mathématiques. La méthodologie employée permet d’obtenir des données globales et d’observer finement les activités des élèves en classe et les pratiques des enseignants

International audience; In the Mathematical Working Space (MWS) model, an epistemological plane and a cognitive plane are introduced with a focus on their interactions related to semiotic, instrumental and discursive dimensions (Figure 1a). The model is devoted to the analysis of mathematical work with, specifically, paradigms guiding and orienting the work. Numerous researches are based on the MWS model and the reader may refer to special issues in journals such as Bolema 30(54) and ZDM-Mathematics Education 48(6) in which an introduction to the model is given in the survey paper. Nevertheless, until now, few studies on modeling tasks have been based on the MWS model and we want to highlight recent researches, in particular coming from PhD studies within our team. Based on the modeling cycle (figure 1b) proposed by Blum and Leiss (2005), we suggest, in the poster, some adaptations which help to understand how MWSs can be used. The whole modeling process is not taken in its whole, and we focus more on how the analysis can be refined, mostly between phases 3 and 5 of the cycle, in relation to activity in different mathematical domains. Figure 1a: MWS diagram; Figure 1b: modeling cycle (Blum & Leiss, 2005) Nechache (2016) suggests describing the modeling work in probability situations, with the MWS framework. She identifies the importance of the theoretical referential of the MWSProba in the constitution of the real model. Then, for the analysis of the mathematical part, she fully uses the MWSProba. In the same way, the MWS model can be used for studying other mathematical domains. Derouet (2016) proposes a similar type of use for the mathematical part, but she associates sub-phases to the stages of the cycle in order to investigate the progress of the modeling process. She isolates a part of the cycle containing "real model" and "real results" that she names pseudo-concrete. It allows her to identify, in a modeling situation related to continuous probability, a work within the MWSProba in various working paradigms. In these studies, the MWS model allows to refine the analysis of the mathematical part by taking into account a first horizontal mathematization followed by a second vertical mathematization allowing to strengthen the mathematical model. Other types of change, or transition, are possible, Thematic Working Group 06 Proceedings of CERME10 1031 like the change of MWS or mathematical domains. In his study in relativist kinematics, Moutet (2016) suggests an extension of the MWS model to take into account a change of matters. He considers a second epistemological plane for physics, and he studies the interactions between these two planes and the cognitive level. In these studies, simulation associated with digital models can also be considered as an important stage of the modeling process. It plays two different roles. The first one is in relation to the development of the real model with a simulation close to the initial situation (urn model or a calculator which proposes rolls of dice or coin, in probability). The second role presupposes a stronger mathematical expertise in the MWS of the domain at stake as, for example, the implementation of an algorithm of dichotomy in analysis. Hence, the use of the MWS framework can enrich and strengthen the analysis of the modeling process based on the study of a cycle (figure 1b) in connection with a first resolution of the problem. It constitutes a first interaction between MWS and the modeling cycle, as a first cycle 1 : The modeling problem has been mathematized and it is possible to identify the epistemological and cognitive components of the MWS in relation to the student's activity and realization in the different domains and paradigms. But we can also, in a more didactic way, think of a second cycle aiming at a better understanding of the model and of the mathematical objects introduced to solve the problem by students. In that case, the modeling task proposed by a teacher aims not only at solving a real problem but more deeply at exploring and understanding the numerous uses of a mathematical notion, enriching the MWS, in particular the theoretical referential. This is what we are developing in a work on progress on the exponential function. Acknowledgment